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Approximate Analytical Solutions of Systems of Pdes by Homotopy Analysis Method

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Approximate Analytical Solutions of Systems of Pdes by Homotopy Analysis Method CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Computers and Mathematics with Applications 55 (2008) 2913–2923 www.elsevier.com/locate/camwa Approximate analytical solutions of systems of PDEs by homotopy analysis method A. Sami Bataineh, M.S.M. Noorani, I. Hashim∗ School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi Selangor, Malaysia Received 24 May 2007; received in revised form 2 November 2007; accepted 17 November 2007 Abstract In this paper, the homotopy analysis method (HAM) is applied to obtain series solutions to linear and nonlinear systems of first- and second-order partial differential equations (PDEs). The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of series solutions. It is shown in particular that the solutions obtained by the variational iteration method (VIM) are only special cases of the HAM solutions. c 2007 Elsevier Ltd. All rights reserved. Keywords: Homotopy analysis method; Systems of PDEs; Nonlinear systems; Variational iteration method; Adomian decomposition method 1. Introduction The homotopy analysis method (HAM) was first proposed by Liao in his Ph.D. thesis [1]. A systematic and clear exposition on HAM is given in [2]. In recent years, this method has been successfully employed to solve many types of nonlinear, homogeneous or nonhomogeneous, equations and systems of equations as well as problems in science and engineering [3–23]. Very recently, Ahmad Bataineh et al. [24,25] presented two modifications of HAM to solve linear and nonlinear ODEs. The HAM contains a certain auxiliary parameter h¯ which provides us with a simple way to adjust and control the convergence region and rate of convergence of the series solution. Moreover, by means of the so-called h¯-curve, it is easy to determine the valid regions of h¯ to gain a convergent series solution. Thus, through HAM, explicit analytic solutions of nonlinear problems are possible. Systems of partial differential equations (PDEs) arise in many scientific models such as the propagation of shallow water waves and the Brusselator model of the chemical reaction-diffusion model. Very recently, Batiha et al. [26] improved Wazwaz’s [27] results on the application of the variational iteration method (VIM) to solve some linear and nonlinear systems of PDEs. In [28], Saha Ray implemented the modified Adomian decomposition method (ADM) for solving the coupled sine-Gordon equation. In this paper, we present an alternative approach based on HAM to approximate the solutions of linear and nonlinear systems of first- and second-order PDEs. Comparisons with the solutions obtained by VIM [27] and ADM [28] shall be made. It is shown in particular that the VIM [27] solutions are just special cases of the HAM solutions. ∗ Corresponding author. E-mail address: ishak [email protected] (I. Hashim). 0898-1221/$ - see front matter c 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2007.11.022 2914 A. Sami Bataineh et al. / Computers and Mathematics with Applications 55 (2008) 2913–2923 2. Basic ideas of HAM We consider the following differential equations, Ni [zi (x, t)] = 0, i = 1, 2,..., n, where Ni are nonlinear operators that represent the whole equations, x and t denote the independent variables and zi (x, t) are unknowns function respectively. By means of generalizing the traditional homotopy method, Liao [2] constructed the so-called zero-order deformation equations (1 − q)L[φi (x, t; q) − zi,0(x, t)] = q h¯ i Ni [φi (x, t; q)], (1) where q ∈ [0, 1] is an embedding parameter, h¯ i are nonzero auxiliary functions, L is an auxiliary linear operator, zi,0(x, t) are initial guesses of zi (x, t) and φi (x, t; q) are unknown functions. It is important to note that, one has great freedom to choose auxiliary objects such as h¯ i and L in HAM. Obviously, when q = 0 and q = 1, both φi (x, t; 0) = zi,0(x, t) and φi (x, t; 1) = zi (x, t), hold. Thus as q increases from 0 to 1, the solutions φi (x, t; q) varies from the initial guesses zi,0(x, t) to the solutions zi (x, t). Expanding φi (x, t; q) in Taylor series with respect to q, one has +∞ X m φi (x, t; q) = zi,0(x, t) + zi,m(x, t)q , (2) m=1 where 1 ∂mφ (x, t; q) = i zi,m m . (3) m! ∂q q=0 If the auxiliary linear operator, the initial guesses, the auxiliary parameters h¯ i , and the auxiliary functions are properly chosen, then the series equation (2) converges at q = 1 and +∞ X φi (x, t; 1) = zi,0(x, t) + zi,m(x, t), m=1 which must be one of solutions of the original nonlinear equations, as proved by Liao [2]. As h¯ i = −1, Eq. (1) becomes (1 − q)L[φi (x, t; q) − zi,0(x, t)] + q Ni [φi (x, t; q)] = 0, (4) which are mostly used in the homotopy-perturbation method [29]. According to (3), the governing equations can be deduced from the zero-order deformation equations (1). Define the vectors Ezi,n = {zi,0(x, t), zi,1(x, t), . , zi,n(x, t)}. Differentiating (1) m times with respect to the embedding parameter q and then setting q = 0 and finally dividing them by m!, we have the so-called mth-order deformation equations L[zi,m(x, t) − χm zi,m−1(x, t)] = h¯ i Ri,m(Ezi,m−1), (5) where 1 ∂m−1 N [φ (x, t; q)] R (Ez ) = i i , i,m i,m−1 m−1 (6) (m − 1)! ∂q q=0 and 0, m ≤ 1, χ = m 1, m > 1. A. Sami Bataineh et al. / Computers and Mathematics with Applications 55 (2008) 2913–2923 2915 It should be emphasized that zi,m(x, t) (m ≥ 1) is governed by the linear equation (5) with the linear boundary conditions that come from the original problem, which can be easily solved by symbolic computation softwares such as Maple and Mathematica. 3. Applications We will apply the HAM to linear and nonlinear systems of PDEs to illustrate the strength of the method and to establish exact and/or approximate solutions for these problems. Comparisons with the VIM [27] and ADM [28] shall be made. 3.1. Homogeneous linear system First we present the analytical solutions for the linear homogeneous system of PDEs: ut − vx + (u + v) = 0, (7) vt − ux + (u + v) = 0, (8) subject to the initial conditions u(x, 0) = sinh x, v(x, 0) = cosh x. (9) To solve system (7)–(9) by means of homotopy analysis method HAM, we choose the initial approximations u0(x, t) = sinh x, v0(x, t) = cosh x, and the linear operator ∂φi (x, t; q) L[φi (x, t; q)] = , i = 1, 2, (10) ∂t with the property L[ci ] = 0, (11) where ci (i = 1, 2) are integral constants. Furthermore, systems (7) and (8) suggest that we define a system of nonlinear operators as ∂φ1(x, t; q) ∂φ2(x, t; q) N [φi (x, t; q)] = − + φ (x, t; q) + φ (x, t; q), 1 ∂t ∂x 1 2 ∂φ2(x, t; q) ∂φ1(x, t; q) N [φi (x, t; q)] = − + φ (x, t; q) + φ (x, t; q). 2 ∂t ∂x 1 2 Using the above definition, we construct the zeroth-order deformation equations (1 − q)L φi (x, t; q) − zi,0(x, t) = q h¯ i Ni [φi (x, t; q)] , i = 1, 2. (12) Obviously, when q = 0 and q = 1, φ1(x, t; 0) = z1,0(x, t) = u0(x, t), φ1(x, t; 1) = u(x, t), φ2(x, t; 0) = z2,0(x, t) = v0(x, t), φ2(x, t; 1) = v(x, t). Therefore, as the embedding parameter q increases from 0 to 1, φi (x, t; q) varies from the initial guess zi,0(x, t) to the solution zi (x, t) for i = 1, 2. Expanding φi (x, t; q) in Taylor series with respect to q one has +∞ X m φi (x, t; q) = zi,0(x, t) + zi,m(x, t)q , m=1 where 1 ∂mφ (x, t; q) = i zi,m(x, t) m . m! ∂q q=0 2916 A. Sami Bataineh et al. / Computers and Mathematics with Applications 55 (2008) 2913–2923 If the auxiliary linear operator, the initial guesses and the auxiliary parameters h¯ i are properly chosen, the above series is convergent at q = 1, then one has +∞ X u(x, t) = z1,0(x, t) + z1,m(x, t), m=1 +∞ X u(x, t) = z2,0(x, t) + z2,m(x, t), m=1 which must be one of the solutions of the original nonlinear equations, as proved by Liao [2]. Now we define the vector Ezi,n = {zi,0(x, t), zi,1(x, t), . , zi,n(x, t)}. So the mth-order deformation equations is L zi,m(x, t) − χm zi,m−1(x, t) = h¯ i Ri,m Ezi,m−1 , (13) with the initial conditions zi,m(x, 0) = 0, (14) where R1,m(Ezi,m−1) = (z1,m−1)t − (z2,m−1)x + z1,m−1 + z2,m−1, R2,m(Ezi,m−1) = (z2,m−1)t − (z1,m−1)x + z1,m−1 + z2,m−1. Now, the solution of the mth-order deformation equation (13) for m ≥ 1 becomes Z t zi,m(x, t) = χm zi,m−1(x, t) + h¯ i Ri,m(Ezi,m−1)dτ + ci , (15) 0 where the integration constants ci (i = 1, 2) are determined by the initial conditions (14).
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